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Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  25 October 2016

Lei Wei
Lei Wei*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, People's Republic of China (wlxznu@163.com)
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Extract

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

Keywords

Hardy potentialsymmetryuniquenessasymptotic behavior

MSC classification

Secondary: 35B40: Asymptotic behavior of solutions 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 6 , December 2016 , pp. 1243 - 1263
Copyright
Copyright © Royal Society of Edinburgh 2016 

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